So what you would like to do today is we would finalize a little bit our discussion of linear

parabolic equations where we have now seen existence, a proof of existence for weak solutions

and quite weak assumptions in particular also for the right hand side only requiring that

then of course L2 in time but only H minus 1 in space function allowing for concentrated

sources. So as you know already one important property of elliptic and parabolic equation is

the maximum principle. The maximum principle comes in various versions very weak formulations up to

very strong formulation the most strong formulation which holds also true in the parabolic sense but

only for classical solutions is that if a maximum is attained in the aetheria the solution is

necessarily constant, constant up to this point so to speak in the parabolic cylinder up to this

time level and then there are all that would be the strong maximum principle then there are

weaker forms only saying the maximum is attained at the boundary and as a consequence there are

comparison principles which ensure uniqueness and most weak formulation which also holds true for

this general solutions which we have is the following. So we are looking at a solution in

the most weak sense so L2 in time H1 in space should be a solution of let's say our standard

equation but that is not so important so this minus Laplace of U can also substitute it by a

general elliptic operator the only thing one has to deal with take care of is the zero order term

which appears in it so there the corresponding signs have to hold but for example first order

terms convective terms there's no restriction whatsoever but we formulate it here in this form

first of all for the homogeneous case I will say something for the inhomogeneous case in a second

and okay then we have initial data and we have boundary data which are here supposed to be

Dirichlet boundary data that's strange okay delta omega t okay so the assumptions the regularity

assumptions are supposed to be okay U0 being an L2 function this was the most general situation

which we had the G is supposed to be an L2 function so the G to be understood as the

prolongation is this necessarily existing prolongation of the boundary data because

the G has to be a trace in space so in the prolongation it should be an H1 function that's

the one thing and we have seen what we further at least need to do the way we did to prove namely

so to speak subtracting the G from the problem we also need something on the time derivative of the

G so we need something on H1 in L in time but here we only need very weak things namely to be a dual

element and being an H minus one of omega so that's the standard assumption and this very weak

formulation of the of the maximum principle is the following if there is a bound for the initial

data of course that means it's actually an L infinity does not mean it's an infinity function

because it's only a one-sided statement here but it goes in the direction the same thing for the

Dirichlet data so we had then the consequence is that the whole solution U is also less or equal

to this constant on Qt so that would be a result which follows from all the other versions of the

maximum principle I have stated so in this sense is a very weak formulation but it already requires

weak assumptions so what kind of so first of all the same thing of course could be also stated as

a minimum principle because then we just look at minus U as this is possible because we have

here a homogeneous equation if you would have an inhomogeneous equation the question is how in which

way the sign of the right hand side and the sign of the source comes in and the proof basically the

proof is cut just comes by at least formally the proof is very simple to make it rigorous is not

not that simple the formal proof is we just test so we just test with a specific function namely

with a test function U minus M plus what we would like to see is that U is less or equal to M that

is we would like to see that the plus part of this function is zero we'd like to prove that this

phi is zero that is to show is phi is equal to zero and if we do the question now is a little

bit how to do derivatives with this how to compute derivatives of this function here because as you

will imagine there is so to speak the set U equals to M and at the set U equals to M typically this

function will be continuous but not differentiable so we can only deal with weak derivatives and we

have to know how to compute the weak derivatives of this function and how to compute the weak

derivatives this is said this is that there is something which is called Stumpakia's lemma

to see Stumpakia's lemma that tells us that tells us that if we would like and that is for the